Track / Stage 3 / 43 of 260 #43 of 1964

Problem 43

AMC 10/12, early questions

Geometry Difficulty 3.1 Find the answer

The ratio of the areas of two concentric circles is $1: 3$ . If the radius of the smaller is $r$ , then the difference between the
radii is best approximated by:

Pick one

Official solution

Observe that there are 3 feet in 1 yard, 9 square feet in 1 square yard, and 27 cubic feet in 1 cubic yard.
This means that if the ratio is $x$ in 1D, it is $x^2$ in 2D, and it is $x^3$ in 3D.
We can apply this thinking into our current question, which would have $1:3$ in the 2D-plane. Since the radius is 1D, we would have to square root both sides to get $1:\sqrt{3}$
If we plug in $r$ for the smaller radius, we get $r:\sqrt{3}r$ The difference between the two sides would be (approximately) $1.732r - r = \textbf{(D)} 0.73r$

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.